Synopsis This note takes an exploratory approach to define and then visualize the notion of polygonal number similarity between pairs of k-gonal numbers P-k(alpha n) and P-k(alpha n), where alpha is an integer scale factor of 2 or greater.
In his article "Memoire sur les Integrales Definies, prises entre des limites imaginaires" of 1825, Cauchy proves that the integral of a complex function f defined on complex numbers is independent of the path of integration if the function f remains finite and continuous along the chosen path. In contemporary literature that analyzes the mathematics of Cauchy, the specific mathematical process that allowed Cauchy to prove this result is not explicitly presented. The purpose of this manuscript is to communicate a symbolic-algebraic derivation that justifies the narrative argument used by Cauchy to support his proof, thus hopefully filling this gap. We do this via a careful reading of some of the other original works of Cauchy. Our reconstruction of Cauchy's mathematical process is framed within a research project in mathematics education that seeks to recover how historical subjects did mathematics in what we now call complex analysis. We hope that our work can serve as an example of how research in mathematics education can recover different ways in which historical subjects did mathematics, ways that are not being explicitly addressed in contemporary literature, but that can serve as a starting point for didactic innovations in complex analysis.
The history of titration dates back to Archimedes who established that an object submerged in a liquid displaces an amount of liquid whose weight is equal to the buoyant force acting on the object. Since then, many scientists and engineers have tried to optimize his approach or devise new instruments for titration purposes. Omar Khayyam (1048-1123), in addition to designing a hydro-static balance, developed mathematical approaches for titration of binary alloys, that is, alloys made up of only two elements. Here we explore the different versions of Khayyam's work on titration given by various sources. We also compare the accuracy of the results of Khayyam's innovative methods with today's real values. The results show incredible precision in the outcomes of Khayyam's methods.
In this article, I reflect on my twenty years as a high school math teacher. I describe four distinct phases of my career in five-year increments. I aim to make public both what makes teaching high school students rewarding and what makes it difficult. Also, my writing illuminates how indelible the first five years were to my development as a teacher. I hope that this brief memoir helps early-career teachers see past the tumultuous beginnings while also inspiring veteran teachers to reflect on their own careers.
Math circles are a common form of outreach usually run by professional mathematicians as enrichment for K-12 students or teachers. As a setting to stimulate interest and develop positive socialization around mathematical practice, a regular math circle can also be an effective bridge to higher education in carceral settings for both facilitators and participants. This article describes my journey through starting and running math circles in county jails, with a view towards the expansion of mathematics education as another tool for the dismantling of mass incarceration.
If mathematics is a language, then African American Vernacular English (AAVE) could hold the key to a fascinating understanding of the evolution of algebra from human speech in North America. The use of the pronoun something (sumthin) to depict an unknown amount of money that varies implies that algebra developed as a logical necessity in human communication. Exploring this phenomenon and leveraging it to demonstrate the concept of algebra could enhance the comprehension of the subject among speakers of AAVE.
Starting from the definition of a harmonic function series, we define a new series which we call the perturbed harmonic function series. We explore the relationship of the new series with the Weierstrass and Riemann fractal functions as well as its convergence and differentiability properties. We then illustrate the potential of the harmonic functions in generating complex figures that sometimes resemble natural objects, with explicit numerical examples.
We find Anne and Toby, as is their nightly ritual, studying in the college library after a tediously long day of classes at Salado College, a small private college located in a sleepy village in the heart of Texas. Toby is a first-time freshman with an undeclared major. He has a strong intuition. Anne is a junior who has made a late decision to major in mathematics. She switched from politics to mathematics while on a study-abroad program. She had a couple of classes on proofs, one required of all students and one required of mathematics majors. She is compulsively analytic and careful with language to a fault. The classes did not have a prerequisite other than emotional maturity and a promise of some mathematical talent. Anne grew up on a farm about thirty miles away. Her parents grew cotton and corn, now mostly corn. Toby grew up on the outskirts of a small town. One could say that they were somewhat sheltered in their growing up days. Both were honors students at their respective high schools. Although at different levels of preparation, Anne and Toby are in Dr. Bradford's first calculus course together. The contention she has concerning empty sets may seem strange to the reader considering her exposure to doing rigorous proofs. For philosophical reasons, all sets in both of her proof classes had at least one element, a throwback to an earlier time when empty sets were not mentioned. Sometimes, the concept was thought to be taboo. Her professor for these two courses had even been challenged during his doctoral defense for not considering empty sets; such was the controversy over a matter strictly of philosophical concern. Of course, he allows empty sets in his more advanced courses so that his students have a background compatible with future colleagues. Anne and Toby have some pressing homework for Dr. Bradford's calculus class. He has decided to give them some enrichment, a subject that has fallen out of favor in contemporary calculus courses, but was a subject of serious study in calculus courses decades ago; the completeness of the reals, sometimes called the continuum. They have been working hard to understand Dr. Bradford's lecture on the topic, specifically, an open-ended, challenge problem he has given them, which is to prove that "gaps" cannot exist on the number line. This is a term Dedekind used in his 1872 seminal paper, Continuity and Irrational Numbers /5]. Dr. Bradford gave a lecture to set up his class, covering concepts that might prove useful to the students. He also told them that they could use the fact that the numbers satisfy the S1S2 property given below, a substitute for the numbers being connected, a new concept for his students: If S1 and S2 are two sets of points, each containing at least one point, such that each point is in one or the other, and every point of S1 is to the left of every point of S2, then S1 has a last point or S2 has a first point, but not both. He covered axioms systems, specifically, the axiom system for linear point set theory where "point" is left undefined. The axiom system contained a total of four axioms although the students only need the S1S2 property or axiom. One thing the reader will notice is the use of "sensory referents." See the last page of /10] for the full description of this concept; however, I encourage the reader the thoughtfully read the entire paper. You may find a perspective that is new to you.
We model prejudice in the context of an iterated prisoner's dilemma tournament. Prejudice here is defined as the inability to distinguish amongst members of an identifiable group. We run a computer simulation where agents of one of two groups with defined strategies are randomly matched against each other. The agents either cooperate with or defect against the agent they are matched with. Our focus is on agents who play an unprejudiced version of the Tit-for-Tat strategy (cooperate with any player in the first encounter, and then apply the same strategy [defect or cooperate] that the opposing player played against them in the last encounter) and a prejudiced strategy (cooperate with any member of the other group in the first encounter with a member of the other group, and then apply the same strategy [defect or cooperate] that any member of that group played against them in the last encounter with a member of that group). We find that even though agents are initially inclined to give members of another group the benefit of the doubt when interacting with them, the introduction of very minor levels of defection, real or perceived, can lead to universal defection against members of the other group.
